How can I get Blender to snap vert using grab command?
The problem is that you have proportional editing on, press O, or the blue dot at the bottom of the viewport to turn it off (see circled in red) 
You also might have forgotten to turn auto merge on (circled in blue), as you seem to have 4 vertices overlapping. To remove them, select everything (A) and use the Remove Doubles function (Press space and search up Remove Doubles).